91Ó°ÊÓ

The Data Encryption Standard (DES) is a symmetric-key algorithm for the encryption of digital data. Although it’s considered to be outdated today due to its 56-bit key size, it was widely used for document security and is an important part in the history of cryptography. DES operates on 64-bit blocks of data, using a shared secret key for both encryption and decryption. The 56-bit key generates 16 subkeys through a series of permutations and substitutions, known as the Feistel structure, during the encryption process. Understanding the structure of DES is crucial for appreciating why even a relatively small key size like 56 bits was initially considered secure.

In encryption, DES transforms plaintext into ciphertext using complex operations involving the key. One distinctive feature of DES is its use of the same algorithm for both encryption and decryption, a trait which comes from its symmetric property where a single key is used. This made it practical for a wide range of applications until the emergence of methods to compromise its security, such as exhaustive key searches, which take advantage of its smaller key space.

An exhaustive key search, also known as brute-force attack, is a cryptanalysis technique in which every possible key within the defined space is systematically tested until the correct one is found. In the case of DES, with its 56-bit key size, there are a total of \(2^{56}\) possible keys. On average, one has to test half the key space to find the correct key, making it \(2^{55}\) attempts for DES. While this number may seem large, advances in computing power have made exhaustive searches feasible for smaller key sizes such as the one used by DES.

The concept operates under the principle that given enough time and computational resources, any encrypted message can be decrypted. However, with the increase in key size, such as in AES (Advanced Encryption Standard) with a 128-bit key or larger, the exhaustive key search becomes impractical due to the astronomical number of possible keys, a clear indication why larger key sizes are an essential part of modern encryption algorithms.

Parallel processing dramatically reduces the time required for significant computational tasks such as an exhaustive key search. It involves the simultaneous use of multiple processors to execute different operations concurrently, markedly increasing computing power and speed. The core idea is to split a large computational burden into smaller pieces and solving these pieces side by side, rather than sequentially.

Using \(2^{14}\) processors in parallel, as proposed in the homework exercise, divides the task among them. Instead of one processor taking on the entire workload, it is shared, which significantly cuts down the time necessary to perform the exhaustive search. This use of parallelism is a common approach in modern cryptanalysis, as it exploits the potential of simultaneous operations to handle complex problems more effectively than a single processor could.

Cryptography computation time refers to the amount of time it takes to perform cryptographic operations such as encryption, decryption, or in this case, an exhaustive key search. The time depends on various factors, including the complexity of the algorithm, the size of the key space, and the power of the computational hardware used.

For DES with a 56-bit key, as outlined in the exercise, using one processor with the ability to perform a DES operation in \(2^{-26}\) seconds would result in significant computation time. However, employing parallel processing by using \(2^{14}\) processors can reduce the computation time substantially. As highlighted, the computation time required for the exhaustive search on a single processor was calculated as \(2^{29}/3600\) hours. With \(2^{14}\) processors, each doing a portion of the work, the task is completed in as little as \(2^{15}/3600\) hours—a dramatic decrease that illustrates the power of parallel computing in the realm of cryptography.